Problem: If the integer $k^{}_{}$ is added to each of the numbers $36^{}_{}$, $300^{}_{}$, and $596^{}_{}$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k^{}_{}$.
Explanation: From the given information, we have that the three numbers $\sqrt{36+k}, \; \sqrt{300+k}, \; \sqrt{596+k}$ form an arithmetic progression, in that order. Therefore, we have \[2\sqrt{300+k} = \sqrt{36+k} + \sqrt{596+k}.\]Squaring both sides of the equation, we get \[4(300+k) = (36+k) + 2\sqrt{(36+k)(596+k)} + (596+k)\]or \[568 + 2k = 2\sqrt{(36+k)(596+k)}.\]Dividing by $2$ and then squaring again, we have \[(284+k)^2 = (36+k)(596+k),\]or \[284^2 + 2 \cdot 284k + k^2 = 36 \cdot 596 + 632k + k^2.\]Thus, \[k = \frac{284^2 - 36 \cdot 596}{632 - 2\cdot 284} = \frac{284^2 - 36 \cdot 596}{64} = \boxed{925}.\]